jmc

Adding the whole numbers gives $2+3+4+5=14$. Adding the fractions gives $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{30}{60}+\frac{20}{60}+\frac{15}{60}+\frac{12}{60}=\frac{77}{60}$. This will create one more whole, and a fraction that is less than $1$. Thus, the smallest whole number that is less than $15$ plus some fractional part is $16$, and the answer is $\boxed{16}$.